The quantum point-mass pendulum

Cook, Gayle; Zaidins, C.S.

doi:10.1119/1.14640

Cited by 23 publications

(27 citation statements)

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“…In reality, if perfect preparation was possible, thermal fluctuations of the pendulum would still perturb the pendulum from the metastable orientation and lead to oscillation. Even at zero temperature, unavoidable quantum fluctuations would lead to evolution 1,2 . Although mechanical pendulums operating at the quantum limit are currently unavailable in the lab, it is possible to study quantum many-body systems that have similar dynamical behaviour [3][4][5] .…”

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Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

et al. 2012

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A pendulum prepared perfectly inverted and motionless is a prototype of unstable equilibrium and corresponds to an unstable hyperbolic fixed point in the dynamical phase space. Here, we measure the non-equilibrium dynamics of a spin-1 Bose-Einstein condensate initialized as a minimum uncertainty spin-nematic state to a hyperbolic fixed point of the phase space. Quantum fluctuations lead to non-linear spin evolution along a separatrix and non-Gaussian probability distributions that are measured to be in good agreement with exact quantum calculations up to 0.25 s. At longer times, atomic loss due to the finite lifetime of the condensate leads to larger spin oscillation amplitudes, as orbits depart from the separatrix. This demonstrates how decoherence of a many-body system can result in apparent coherent behaviour. This experiment provides new avenues for studying macroscopic spin systems in the quantum limit and for investigations of important topics in non-equilibrium quantum dynamics.

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confidence: 99%

Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

et al. 2012

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“…First tackled by Condon in 1928 [1] in its planar variety, the quantum pendulum has since turned up in a number of research areas of atomic, molecular and optical physics, ranging from spectroscopy to the stereodynamics of molecular collisions to the manipulation of matter by external electric, magnetic and optical fields. Although both the planar and the full-fledged 3D spherical pendular varieties possess analytic asymptotic states [2][3][4][5][6][7][8][9][10], the planar case has been explored with particular tenacity [11][12][13][14][15][16][17][18][19], apparently because of its prototypical character, dwarfed only by that of few other systems such as of the harmonic oscillator.…”

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Supersymmetry and eigensurface topology of the planar quantum pendulum

Schmidt

Friedrich

2014

Front. Physics

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We make use of supersymmetric quantum mechanics (SUSY QM) to find three sets of conditions under which the problem of a planar quantum pendulum becomes analytically solvable. The analytic forms of the pendulum's eigenfuntions make it possible to find analytic expressions for observables of interest, such as the expectation values of the angular momentum squared and of the orientation and alignment cosines as well as of the eigenenergy. Furthermore, we find that the topology of the intersections of the pendulum's eigenenergy surfaces can be characterized by a single integer index whose values correspond to the sets of conditions under which the analytic solutions to the quantum pendulum problem exist

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“…The lack of a true divergence is understandable given the quantized nature of the energy eigenvalues and has been discussed in terms of the explicit time evolution of wave packet solutions as described in Ref. [7], which also outlines simple uncertainty principle arguments.…”

Section: Classical Periodicity Revival and Superrevival Times Fomentioning

confidence: 99%

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

Doncheski

Robinett

2003

Annals of Physics

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The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low-and high-energy limits respectively. The energy variation of the classical periodicity (τ ) is also dramatic, having the special limiting case of τ → ∞ at the 'top' of the classical motion (i.e. the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to 4th order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the 'top' in energy, from either above or below, the revival times decrease from their low-and high-energy values until very near the separatrix where they increase to a large, but finite value.

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The quantum point-mass pendulum

Abstract: The question of how long it takes an inverted point-mass pendulum to fall according to classical, semiclassical, and quantum theories is examined. The semiclassical analysis is based on the uncertainty introduced by measurement.

Cited by 23 publications

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Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

Supersymmetry and eigensurface topology of the planar quantum pendulum

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

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